In these lessons, we learn how to use the double angle formulas and the half-angle formulas to solve trigonometric equations and to prove trigonometric identities. We will also prove the double angle formulas and the half-angle formulas.

**Related Pages**

Lessons On Trigonometry

Trigonometric Identities

Trigonometry Worksheets

The following figure gives the Double-Angle Formulas and Half-Angle Formulas. Scroll down the page for more examples and solutions of how to use, derive and proof the Double-Angle Formulas and Half-Angle Formulas.

**Example:**

Given , find

a) sin 2θ

b)

**Solution:**

a) sin 2θ = 2 sin θ cos θ

Using Double Angle Identities to Solve Equations, Example 1.

This video uses some double angle identities for sine and/or cosine to solve some equations.

**Example:**

cos(4x) − 3cos(2x) = 4

**Using Double Angle Identities to Solve Equations, Example 2**

**Example:**

sin(2t) = sin(t)

**Using Double Angle Identities to Solve Equations, Example 3**

**Example:**

sin(2t) + 4sin(t) + 2cos(t) = −4

Trigonometry Identities - Double Angles (1)

**Example:**

(1 − cos 2x)/sin 2x = tan x

**Trigonometry Identities - Double Angle (2)**

**Example:**

Prove tan x + cos x = 2 cosec 2x

**Double-Angle Formulas**

A few examples that use double-angle formulas from trigonometry.

**Example:**

- If cos x = 1/√10 with x in quadrant IV, find sin 2x
- Graph y = 4 - 8 sin
^{2}x - Verify sin60° = 2sin30°cos30°

Trigonometry - Proof of the Double Angle Formulae from the Addition Formulae.

**How to derive the double angle identities for sine and cosine?**

**Example:**

If cosθ = -4/5 and sinθ > 0, find

a) sin2θ

b) cos2θ

Half-Angle Formulas

A few problems involving half-angle formulas in trigonometry.

**Example:**

Let sinA = 4/5 with A in Quadrant II. Find

- sin A/2
- cos 2A
- sec 2A

**Half Angle Identities to Evaluate Trigonometric Expressions, Example 1**

This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.

**Example:**

Find the exact value of sin(22.5°)

**Half Angle Identities to Evaluate Trigonometric Expressions, Example 2**

This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.

**Example:**

Find the exact value of tan(105°)

**Half Angle Identities to Evaluate Trigonometric Expressions, Example 3**

This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.

Find sin(a/2) if cos a = 3/5 for 0° ≤ a ≤ 90°

This video derives the half angle trigonometry identities for cosine, sine and tangent.

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